L11a451

L11a450

L11a452

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Link Presentations

[edit Notes on L11a451's Link Presentations]

Planar diagram presentation	X6172 X14,3,15,4 X20,10,21,9 X16,8,17,7 X18,12,19,11 X22,16,13,15 X12,18,5,17 X10,20,11,19 X8,22,9,21 X2536 X4,13,1,14
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -9, 3, -8, 5, -7}, {11, -2, 6, -4, 7, -5, 8, -3, 9, -6}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-t(1)t(2)^{3}+t(1)t(3)t(2)^{3}-2t(3)t(2)^{3}+t(2)^{3}+t(1)t(3)^{2}t(2)^{2}-2t(3)^{2}t(2)^{2}+2t(1)t(2)^{2}-3t(1)t(3)t(2)^{2}+3t(3)t(2)^{2}-t(2)^{2}+t(1)t(3)^{3}t(2)-2t(3)^{3}t(2)-3t(1)t(3)^{2}t(2)+3t(3)^{2}t(2)+2t(1)t(3)t(2)-t(3)t(2)-t(1)t(3)^{3}+t(3)^{3}+2t(1)t(3)^{2}-t(3)^{2}}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{7}+3q^{6}-4q^{5}+7q^{4}+q^{-4}-8q^{3}-2q^{-3}+11q^{2}+5q^{-2}-10q-7q^{-1}+9$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{4}-2z^{2}a^{2}+z^{4}-z^{2}+z^{-2}+z^{4}a^{-2}-2z^{2}a^{-2}-2a^{-2}z^{-2}-3a^{-2}+z^{4}a^{-4}+z^{2}a^{-4}+a^{-4}z^{-2}+2a^{-4}-z^{2}a^{-6}$ (db)
Kauffman polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^{10} a^{-2} +z^{10} a^{-4} +2 z^9 a^{-1} +5 z^9 a^{-3} +3 z^9 a^{-5} +2 z^8 a^{-2} +2 z^8 a^{-4} +3 z^8 a^{-6} +3 z^8+3 a z^7-z^7 a^{-1} -17 z^7 a^{-3} -12 z^7 a^{-5} +z^7 a^{-7} +3 a^2 z^6-16 z^6 a^{-2} -22 z^6 a^{-4} -14 z^6 a^{-6} -5 z^6+2 a^3 z^5-2 a z^5-7 z^5 a^{-1} +12 z^5 a^{-3} +11 z^5 a^{-5} -4 z^5 a^{-7} +a^4 z^4-3 a^2 z^4+29 z^4 a^{-2} +35 z^4 a^{-4} +18 z^4 a^{-6} +8 z^4-2 a^3 z^3+14 z^3 a^{-1} +8 z^3 a^{-3} -z^3 a^{-5} +3 z^3 a^{-7} -2 a^4 z^2+a^2 z^2-28 z^2 a^{-2} -23 z^2 a^{-4} -7 z^2 a^{-6} -9 z^2-11 z a^{-1} -11 z a^{-3} +a^4+13 a^{-2} +8 a^{-4} +5+2 a^{-1} z^{-1} +2 a^{-3} z^{-1} -2 a^{-2} z^{-2} - a^{-4} z^{-2} - z^{-2} } (db)

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).

\   r    \    j   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 15                       1 -1 13                     2   2 11                   2 1   -1 9                 5 2     3 7               5 4       -1 5             6 3         3 3           4 5           1 1         5 6             -1 -1       3 5               2 -3     2 4                 -2 -5     3                   3 -7 1 2                     -1 -9 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=-1} $i=1$ $r=-4$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-1$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=0$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{5}$ $r=1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=2$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=4$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{5}$ $r=5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=7$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Link Page master template (intermediate). See/edit the Link_Splice_Base (expert). Back to the top.

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